Stress and strain ratios (isotropic material)

If we now restrict the analysis to isotropic materials, where identical properties will be measured in all directions, we may assume from symmetry that the strains in the width and thickness directions will be equal in magnitude and hence, from dε1 + dε2 + dε3 = 0,

dε2 = dε3 = −(1/2)dε1

(In the previous chapter we considered the case in which the material was anisotropic where dε2 = Rdε3 and the R-value was not unity. We can develop a general theory for anisotropic deformation, but this is not necessary at this stage.) We may summarize the tensile test process for an isotropic material in terms of the strain increments and stresses in the following manner:

dε1 =dl/l ; dε2 = −(1/2)dε1 ; dε3 = −(1/2)dε1
and

σ1 =P/A ; σ2 = 0 ; σ3 = 0